In a series of recent papers, Martel and Merle solved the long-standing open problem on the existence of blow up solutions in the energy space for the critical generalized Korteweg- de Vries equation. Martel and Merle introduced new tools to study the nonlinear dynamics close to a solitary wave solution. The aim of the talk is to discuss the main ideas developed by Martel-Merle, together with a presentation of previously known closely related results.
We survey some recent results for the KP-II equation. We also give an idea for treating the “bad frequency interactions” of the bilinear estimates in the Fourier transform restriction spaces related to the KP-I equation.
We prove the existence and the invariance of a Gibbs measure associated to the defocusing sub-quintic Nonlinear Schrödinger equations on the disc of the plane . We also prove an estimate giving some intuition to what may happen in dimensions.
Inspired by the work of Zhidkov on the KdV equation, we perform a construction of weighted Gaussian measures associated to the higher order conservation laws of the Benjamin-Ono equation. The resulting measures are supported by Sobolev spaces of increasing regularity. We also prove a property on the support of these measures leading to the conjecture that they are indeed invariant by the flow of the Benjamin-Ono equation.
The purpose of this article is to introduce for dispersive partial differential equations with random initial data, the notion of well-posedness (in the Hadamard-probabilistic sense). We restrict the study to one of the simplest examples of such equations: the periodic cubic semi-linear wave equation. Our contributions in this work are twofold: first we break the algebraic rigidity involved in our previous works and allow much more general randomizations (general infinite product measures v.s. Gibbs...
We prove Strichartz estimates with fractional loss of derivatives for the Schrödinger equation on any riemannian compact manifold. As a consequence we infer global existence results for the Cauchy problem of nonlinear Schrödinger equations on surfaces in the case of defocusing polynomial nonlinearities, and on three-manifolds in the case of quadratic nonlinearities. We also discuss the optimality of these Strichartz estimates on spheres.
We summarize the main ideas in a series of papers ([], [], [], []) devoted to the construction of invariant measures and to the long-time behavior of solutions of the periodic Benjamin-Ono equation.
In this article, we first present the construction of Gibbs measures associated to nonlinear Schrödinger equations with harmonic potential. Then we show that the corresponding Cauchy problem is globally well-posed for rough initial conditions in a statistical set (the support of the measures). Finally, we prove that the Gibbs measures are indeed invariant by the flow of the equation. As a byproduct of our analysis, we give a global well-posedness and scattering result for the critical and super-critical...
On étudie l’équation de Schrödinger non linéaire sur les variétés de dimension . On démontre l’existence globale dans pour les non linéarités sous-quintiques. Un élément essentiel de la preuve est une estimation multilinéaire du produit de plusieurs fonctions propres du laplacien sur une variété compacte.
We give local and global well-posedness results for a system of two
Kadomtsev-Petviashvili (KP) equations derived by R. Grimshaw and Y. Zhu
to model the oblique interaction of weakly nonlinear, two dimensional,
long internal waves in shallow fluids.
We also prove a smoothing effect for the amplitudes of the interacting waves.
We use the Fourier transform restriction norms introduced by J. Bourgain
and the Strichartz estimates for the linear KP group. Finally
we extend the result of [3] for lower...
This text aims to describe results of the authors on the long time behavior of NLS on product spaces with a particular emphasis on the existence of solutions with growing higher Sobolev norms.
Cet exposé concerne l’approximation faiblement non-linéaire de problèmes aux limites invariants par changement d’échelles.
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